Projective Modules and Polynomials over the Hurwitz Quaternions

A theorem of Sheshadri (Proc. Nat. Acad. Sci. USA 44 (1958) 456–458) shows that, when Λ is a commutative principal ideal domain, finitely generated projective modules over the polynomial ring Λ[t] are all free. The ring Γ of Hurwitz quaternions, that is, the unique maximal order in the ring of rational quaternions, is the simplest example of a non-commutative principal ideal domain which is not a division ring. In contrast to the commutative case, we show that Γ[t] has infinitely many isomorphically distinct projective modules; these are stably free of rank 1.